author hoelzl Mon Aug 08 14:13:14 2016 +0200 (2016-08-08) changeset 63627 6ddb43c6b711 parent 63626 src/HOL/Multivariate_Analysis/Radon_Nikodym.thy@44ce6b524ff3 child 64283 979cdfdf7a79 permissions -rw-r--r--
rename HOL-Multivariate_Analysis to HOL-Analysis.
```     1 (*  Title:      HOL/Analysis/Radon_Nikodym.thy
```
```     2     Author:     Johannes Hölzl, TU München
```
```     3 *)
```
```     4
```
```     5 section \<open>Radon-Nikod{\'y}m derivative\<close>
```
```     6
```
```     7 theory Radon_Nikodym
```
```     8 imports Bochner_Integration
```
```     9 begin
```
```    10
```
```    11 definition diff_measure :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a measure"
```
```    12 where
```
```    13   "diff_measure M N = measure_of (space M) (sets M) (\<lambda>A. emeasure M A - emeasure N A)"
```
```    14
```
```    15 lemma
```
```    16   shows space_diff_measure[simp]: "space (diff_measure M N) = space M"
```
```    17     and sets_diff_measure[simp]: "sets (diff_measure M N) = sets M"
```
```    18   by (auto simp: diff_measure_def)
```
```    19
```
```    20 lemma emeasure_diff_measure:
```
```    21   assumes fin: "finite_measure M" "finite_measure N" and sets_eq: "sets M = sets N"
```
```    22   assumes pos: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure N A \<le> emeasure M A" and A: "A \<in> sets M"
```
```    23   shows "emeasure (diff_measure M N) A = emeasure M A - emeasure N A" (is "_ = ?\<mu> A")
```
```    24   unfolding diff_measure_def
```
```    25 proof (rule emeasure_measure_of_sigma)
```
```    26   show "sigma_algebra (space M) (sets M)" ..
```
```    27   show "positive (sets M) ?\<mu>"
```
```    28     using pos by (simp add: positive_def)
```
```    29   show "countably_additive (sets M) ?\<mu>"
```
```    30   proof (rule countably_additiveI)
```
```    31     fix A :: "nat \<Rightarrow> _"  assume A: "range A \<subseteq> sets M" and "disjoint_family A"
```
```    32     then have suminf:
```
```    33       "(\<Sum>i. emeasure M (A i)) = emeasure M (\<Union>i. A i)"
```
```    34       "(\<Sum>i. emeasure N (A i)) = emeasure N (\<Union>i. A i)"
```
```    35       by (simp_all add: suminf_emeasure sets_eq)
```
```    36     with A have "(\<Sum>i. emeasure M (A i) - emeasure N (A i)) =
```
```    37       (\<Sum>i. emeasure M (A i)) - (\<Sum>i. emeasure N (A i))"
```
```    38       using fin pos[of "A _"]
```
```    39       by (intro ennreal_suminf_minus)
```
```    40          (auto simp: sets_eq finite_measure.emeasure_eq_measure suminf_emeasure)
```
```    41     then show "(\<Sum>i. emeasure M (A i) - emeasure N (A i)) =
```
```    42       emeasure M (\<Union>i. A i) - emeasure N (\<Union>i. A i) "
```
```    43       by (simp add: suminf)
```
```    44   qed
```
```    45 qed fact
```
```    46
```
```    47 lemma (in sigma_finite_measure) Ex_finite_integrable_function:
```
```    48   "\<exists>h\<in>borel_measurable M. integral\<^sup>N M h \<noteq> \<infinity> \<and> (\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>)"
```
```    49 proof -
```
```    50   obtain A :: "nat \<Rightarrow> 'a set" where
```
```    51     range[measurable]: "range A \<subseteq> sets M" and
```
```    52     space: "(\<Union>i. A i) = space M" and
```
```    53     measure: "\<And>i. emeasure M (A i) \<noteq> \<infinity>" and
```
```    54     disjoint: "disjoint_family A"
```
```    55     using sigma_finite_disjoint by blast
```
```    56   let ?B = "\<lambda>i. 2^Suc i * emeasure M (A i)"
```
```    57   have [measurable]: "\<And>i. A i \<in> sets M"
```
```    58     using range by fastforce+
```
```    59   have "\<forall>i. \<exists>x. 0 < x \<and> x < inverse (?B i)"
```
```    60   proof
```
```    61     fix i show "\<exists>x. 0 < x \<and> x < inverse (?B i)"
```
```    62       using measure[of i]
```
```    63       by (auto intro!: dense simp: ennreal_inverse_positive ennreal_mult_eq_top_iff power_eq_top_ennreal)
```
```    64   qed
```
```    65   from choice[OF this] obtain n where n: "\<And>i. 0 < n i"
```
```    66     "\<And>i. n i < inverse (2^Suc i * emeasure M (A i))" by auto
```
```    67   { fix i have "0 \<le> n i" using n(1)[of i] by auto } note pos = this
```
```    68   let ?h = "\<lambda>x. \<Sum>i. n i * indicator (A i) x"
```
```    69   show ?thesis
```
```    70   proof (safe intro!: bexI[of _ ?h] del: notI)
```
```    71     have "integral\<^sup>N M ?h = (\<Sum>i. n i * emeasure M (A i))" using pos
```
```    72       by (simp add: nn_integral_suminf nn_integral_cmult_indicator)
```
```    73     also have "\<dots> \<le> (\<Sum>i. ennreal ((1/2)^Suc i))"
```
```    74     proof (intro suminf_le allI)
```
```    75       fix N
```
```    76       have "n N * emeasure M (A N) \<le> inverse (2^Suc N * emeasure M (A N)) * emeasure M (A N)"
```
```    77         using n[of N] by (intro mult_right_mono) auto
```
```    78       also have "\<dots> = (1/2)^Suc N * (inverse (emeasure M (A N)) * emeasure M (A N))"
```
```    79         using measure[of N]
```
```    80         by (simp add: ennreal_inverse_power divide_ennreal_def ennreal_inverse_mult
```
```    81                       power_eq_top_ennreal less_top[symmetric] mult_ac
```
```    82                  del: power_Suc)
```
```    83       also have "\<dots> \<le> inverse (ennreal 2) ^ Suc N"
```
```    84         using measure[of N]
```
```    85         by (cases "emeasure M (A N)"; cases "emeasure M (A N) = 0")
```
```    86            (auto simp: inverse_ennreal ennreal_mult[symmetric] divide_ennreal_def simp del: power_Suc)
```
```    87       also have "\<dots> = ennreal (inverse 2 ^ Suc N)"
```
```    88         by (subst ennreal_power[symmetric], simp) (simp add: inverse_ennreal)
```
```    89       finally show "n N * emeasure M (A N) \<le> ennreal ((1/2)^Suc N)"
```
```    90         by simp
```
```    91     qed auto
```
```    92     also have "\<dots> < top"
```
```    93       unfolding less_top[symmetric]
```
```    94       by (rule ennreal_suminf_neq_top)
```
```    95          (auto simp: summable_geometric summable_Suc_iff simp del: power_Suc)
```
```    96     finally show "integral\<^sup>N M ?h \<noteq> \<infinity>"
```
```    97       by (auto simp: top_unique)
```
```    98   next
```
```    99     { fix x assume "x \<in> space M"
```
```   100       then obtain i where "x \<in> A i" using space[symmetric] by auto
```
```   101       with disjoint n have "?h x = n i"
```
```   102         by (auto intro!: suminf_cmult_indicator intro: less_imp_le)
```
```   103       then show "0 < ?h x" and "?h x < \<infinity>" using n[of i] by (auto simp: less_top[symmetric]) }
```
```   104     note pos = this
```
```   105   qed measurable
```
```   106 qed
```
```   107
```
```   108 subsection "Absolutely continuous"
```
```   109
```
```   110 definition absolutely_continuous :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> bool" where
```
```   111   "absolutely_continuous M N \<longleftrightarrow> null_sets M \<subseteq> null_sets N"
```
```   112
```
```   113 lemma absolutely_continuousI_count_space: "absolutely_continuous (count_space A) M"
```
```   114   unfolding absolutely_continuous_def by (auto simp: null_sets_count_space)
```
```   115
```
```   116 lemma absolutely_continuousI_density:
```
```   117   "f \<in> borel_measurable M \<Longrightarrow> absolutely_continuous M (density M f)"
```
```   118   by (force simp add: absolutely_continuous_def null_sets_density_iff dest: AE_not_in)
```
```   119
```
```   120 lemma absolutely_continuousI_point_measure_finite:
```
```   121   "(\<And>x. \<lbrakk> x \<in> A ; f x \<le> 0 \<rbrakk> \<Longrightarrow> g x \<le> 0) \<Longrightarrow> absolutely_continuous (point_measure A f) (point_measure A g)"
```
```   122   unfolding absolutely_continuous_def by (force simp: null_sets_point_measure_iff)
```
```   123
```
```   124 lemma absolutely_continuousD:
```
```   125   "absolutely_continuous M N \<Longrightarrow> A \<in> sets M \<Longrightarrow> emeasure M A = 0 \<Longrightarrow> emeasure N A = 0"
```
```   126   by (auto simp: absolutely_continuous_def null_sets_def)
```
```   127
```
```   128 lemma absolutely_continuous_AE:
```
```   129   assumes sets_eq: "sets M' = sets M"
```
```   130     and "absolutely_continuous M M'" "AE x in M. P x"
```
```   131    shows "AE x in M'. P x"
```
```   132 proof -
```
```   133   from \<open>AE x in M. P x\<close> obtain N where N: "N \<in> null_sets M" "{x\<in>space M. \<not> P x} \<subseteq> N"
```
```   134     unfolding eventually_ae_filter by auto
```
```   135   show "AE x in M'. P x"
```
```   136   proof (rule AE_I')
```
```   137     show "{x\<in>space M'. \<not> P x} \<subseteq> N" using sets_eq_imp_space_eq[OF sets_eq] N(2) by simp
```
```   138     from \<open>absolutely_continuous M M'\<close> show "N \<in> null_sets M'"
```
```   139       using N unfolding absolutely_continuous_def sets_eq null_sets_def by auto
```
```   140   qed
```
```   141 qed
```
```   142
```
```   143 subsection "Existence of the Radon-Nikodym derivative"
```
```   144
```
```   145 lemma (in finite_measure) Radon_Nikodym_finite_measure:
```
```   146   assumes "finite_measure N" and sets_eq[simp]: "sets N = sets M"
```
```   147   assumes "absolutely_continuous M N"
```
```   148   shows "\<exists>f \<in> borel_measurable M. density M f = N"
```
```   149 proof -
```
```   150   interpret N: finite_measure N by fact
```
```   151   define G where "G = {g \<in> borel_measurable M. \<forall>A\<in>sets M. (\<integral>\<^sup>+x. g x * indicator A x \<partial>M) \<le> N A}"
```
```   152   have [measurable_dest]: "f \<in> G \<Longrightarrow> f \<in> borel_measurable M"
```
```   153     and G_D: "\<And>A. f \<in> G \<Longrightarrow> A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+x. f x * indicator A x \<partial>M) \<le> N A" for f
```
```   154     by (auto simp: G_def)
```
```   155   note this[measurable_dest]
```
```   156   have "(\<lambda>x. 0) \<in> G" unfolding G_def by auto
```
```   157   hence "G \<noteq> {}" by auto
```
```   158   { fix f g assume f[measurable]: "f \<in> G" and g[measurable]: "g \<in> G"
```
```   159     have "(\<lambda>x. max (g x) (f x)) \<in> G" (is "?max \<in> G") unfolding G_def
```
```   160     proof safe
```
```   161       let ?A = "{x \<in> space M. f x \<le> g x}"
```
```   162       have "?A \<in> sets M" using f g unfolding G_def by auto
```
```   163       fix A assume [measurable]: "A \<in> sets M"
```
```   164       have union: "((?A \<inter> A) \<union> ((space M - ?A) \<inter> A)) = A"
```
```   165         using sets.sets_into_space[OF \<open>A \<in> sets M\<close>] by auto
```
```   166       have "\<And>x. x \<in> space M \<Longrightarrow> max (g x) (f x) * indicator A x =
```
```   167         g x * indicator (?A \<inter> A) x + f x * indicator ((space M - ?A) \<inter> A) x"
```
```   168         by (auto simp: indicator_def max_def)
```
```   169       hence "(\<integral>\<^sup>+x. max (g x) (f x) * indicator A x \<partial>M) =
```
```   170         (\<integral>\<^sup>+x. g x * indicator (?A \<inter> A) x \<partial>M) +
```
```   171         (\<integral>\<^sup>+x. f x * indicator ((space M - ?A) \<inter> A) x \<partial>M)"
```
```   172         by (auto cong: nn_integral_cong intro!: nn_integral_add)
```
```   173       also have "\<dots> \<le> N (?A \<inter> A) + N ((space M - ?A) \<inter> A)"
```
```   174         using f g unfolding G_def by (auto intro!: add_mono)
```
```   175       also have "\<dots> = N A"
```
```   176         using union by (subst plus_emeasure) auto
```
```   177       finally show "(\<integral>\<^sup>+x. max (g x) (f x) * indicator A x \<partial>M) \<le> N A" .
```
```   178     qed auto }
```
```   179   note max_in_G = this
```
```   180   { fix f assume  "incseq f" and f: "\<And>i. f i \<in> G"
```
```   181     then have [measurable]: "\<And>i. f i \<in> borel_measurable M" by (auto simp: G_def)
```
```   182     have "(\<lambda>x. SUP i. f i x) \<in> G" unfolding G_def
```
```   183     proof safe
```
```   184       show "(\<lambda>x. SUP i. f i x) \<in> borel_measurable M" by measurable
```
```   185     next
```
```   186       fix A assume "A \<in> sets M"
```
```   187       have "(\<integral>\<^sup>+x. (SUP i. f i x) * indicator A x \<partial>M) =
```
```   188         (\<integral>\<^sup>+x. (SUP i. f i x * indicator A x) \<partial>M)"
```
```   189         by (intro nn_integral_cong) (simp split: split_indicator)
```
```   190       also have "\<dots> = (SUP i. (\<integral>\<^sup>+x. f i x * indicator A x \<partial>M))"
```
```   191         using \<open>incseq f\<close> f \<open>A \<in> sets M\<close>
```
```   192         by (intro nn_integral_monotone_convergence_SUP)
```
```   193            (auto simp: G_def incseq_Suc_iff le_fun_def split: split_indicator)
```
```   194       finally show "(\<integral>\<^sup>+x. (SUP i. f i x) * indicator A x \<partial>M) \<le> N A"
```
```   195         using f \<open>A \<in> sets M\<close> by (auto intro!: SUP_least simp: G_D)
```
```   196     qed }
```
```   197   note SUP_in_G = this
```
```   198   let ?y = "SUP g : G. integral\<^sup>N M g"
```
```   199   have y_le: "?y \<le> N (space M)" unfolding G_def
```
```   200   proof (safe intro!: SUP_least)
```
```   201     fix g assume "\<forall>A\<in>sets M. (\<integral>\<^sup>+x. g x * indicator A x \<partial>M) \<le> N A"
```
```   202     from this[THEN bspec, OF sets.top] show "integral\<^sup>N M g \<le> N (space M)"
```
```   203       by (simp cong: nn_integral_cong)
```
```   204   qed
```
```   205   from ennreal_SUP_countable_SUP [OF \<open>G \<noteq> {}\<close>, of "integral\<^sup>N M"] guess ys .. note ys = this
```
```   206   then have "\<forall>n. \<exists>g. g\<in>G \<and> integral\<^sup>N M g = ys n"
```
```   207   proof safe
```
```   208     fix n assume "range ys \<subseteq> integral\<^sup>N M ` G"
```
```   209     hence "ys n \<in> integral\<^sup>N M ` G" by auto
```
```   210     thus "\<exists>g. g\<in>G \<and> integral\<^sup>N M g = ys n" by auto
```
```   211   qed
```
```   212   from choice[OF this] obtain gs where "\<And>i. gs i \<in> G" "\<And>n. integral\<^sup>N M (gs n) = ys n" by auto
```
```   213   hence y_eq: "?y = (SUP i. integral\<^sup>N M (gs i))" using ys by auto
```
```   214   let ?g = "\<lambda>i x. Max ((\<lambda>n. gs n x) ` {..i})"
```
```   215   define f where [abs_def]: "f x = (SUP i. ?g i x)" for x
```
```   216   let ?F = "\<lambda>A x. f x * indicator A x"
```
```   217   have gs_not_empty: "\<And>i x. (\<lambda>n. gs n x) ` {..i} \<noteq> {}" by auto
```
```   218   { fix i have "?g i \<in> G"
```
```   219     proof (induct i)
```
```   220       case 0 thus ?case by simp fact
```
```   221     next
```
```   222       case (Suc i)
```
```   223       with Suc gs_not_empty \<open>gs (Suc i) \<in> G\<close> show ?case
```
```   224         by (auto simp add: atMost_Suc intro!: max_in_G)
```
```   225     qed }
```
```   226   note g_in_G = this
```
```   227   have "incseq ?g" using gs_not_empty
```
```   228     by (auto intro!: incseq_SucI le_funI simp add: atMost_Suc)
```
```   229
```
```   230   from SUP_in_G[OF this g_in_G] have [measurable]: "f \<in> G" unfolding f_def .
```
```   231   then have [measurable]: "f \<in> borel_measurable M" unfolding G_def by auto
```
```   232
```
```   233   have "integral\<^sup>N M f = (SUP i. integral\<^sup>N M (?g i))" unfolding f_def
```
```   234     using g_in_G \<open>incseq ?g\<close> by (auto intro!: nn_integral_monotone_convergence_SUP simp: G_def)
```
```   235   also have "\<dots> = ?y"
```
```   236   proof (rule antisym)
```
```   237     show "(SUP i. integral\<^sup>N M (?g i)) \<le> ?y"
```
```   238       using g_in_G by (auto intro: SUP_mono)
```
```   239     show "?y \<le> (SUP i. integral\<^sup>N M (?g i))" unfolding y_eq
```
```   240       by (auto intro!: SUP_mono nn_integral_mono Max_ge)
```
```   241   qed
```
```   242   finally have int_f_eq_y: "integral\<^sup>N M f = ?y" .
```
```   243
```
```   244   have upper_bound: "\<forall>A\<in>sets M. N A \<le> density M f A"
```
```   245   proof (rule ccontr)
```
```   246     assume "\<not> ?thesis"
```
```   247     then obtain A where A[measurable]: "A \<in> sets M" and f_less_N: "density M f A < N A"
```
```   248       by (auto simp: not_le)
```
```   249     then have pos_A: "0 < M A"
```
```   250       using \<open>absolutely_continuous M N\<close>[THEN absolutely_continuousD, OF A]
```
```   251       by (auto simp: zero_less_iff_neq_zero)
```
```   252
```
```   253     define b where "b = (N A - density M f A) / M A / 2"
```
```   254     with f_less_N pos_A have "0 < b" "b \<noteq> top"
```
```   255       by (auto intro!: diff_gr0_ennreal simp: zero_less_iff_neq_zero diff_eq_0_iff_ennreal ennreal_divide_eq_top_iff)
```
```   256
```
```   257     let ?f = "\<lambda>x. f x + b"
```
```   258     have "nn_integral M f \<noteq> top"
```
```   259       using `f \<in> G`[THEN G_D, of "space M"] by (auto simp: top_unique cong: nn_integral_cong)
```
```   260     with \<open>b \<noteq> top\<close> interpret Mf: finite_measure "density M ?f"
```
```   261       by (intro finite_measureI)
```
```   262          (auto simp: field_simps mult_indicator_subset ennreal_mult_eq_top_iff
```
```   263                      emeasure_density nn_integral_cmult_indicator nn_integral_add
```
```   264                cong: nn_integral_cong)
```
```   265
```
```   266     from unsigned_Hahn_decomposition[of "density M ?f" N A]
```
```   267     obtain Y where [measurable]: "Y \<in> sets M" and [simp]: "Y \<subseteq> A"
```
```   268        and Y1: "\<And>C. C \<in> sets M \<Longrightarrow> C \<subseteq> Y \<Longrightarrow> density M ?f C \<le> N C"
```
```   269        and Y2: "\<And>C. C \<in> sets M \<Longrightarrow> C \<subseteq> A \<Longrightarrow> C \<inter> Y = {} \<Longrightarrow> N C \<le> density M ?f C"
```
```   270        by auto
```
```   271
```
```   272     let ?f' = "\<lambda>x. f x + b * indicator Y x"
```
```   273     have "M Y \<noteq> 0"
```
```   274     proof
```
```   275       assume "M Y = 0"
```
```   276       then have "N Y = 0"
```
```   277         using \<open>absolutely_continuous M N\<close>[THEN absolutely_continuousD, of Y] by auto
```
```   278       then have "N A = N (A - Y)"
```
```   279         by (subst emeasure_Diff) auto
```
```   280       also have "\<dots> \<le> density M ?f (A - Y)"
```
```   281         by (rule Y2) auto
```
```   282       also have "\<dots> \<le> density M ?f A - density M ?f Y"
```
```   283         by (subst emeasure_Diff) auto
```
```   284       also have "\<dots> \<le> density M ?f A - 0"
```
```   285         by (intro ennreal_minus_mono) auto
```
```   286       also have "density M ?f A = b * M A + density M f A"
```
```   287         by (simp add: emeasure_density field_simps mult_indicator_subset nn_integral_add nn_integral_cmult_indicator)
```
```   288       also have "\<dots> < N A"
```
```   289         using f_less_N pos_A
```
```   290         by (cases "density M f A"; cases "M A"; cases "N A")
```
```   291            (auto simp: b_def ennreal_less_iff ennreal_minus divide_ennreal ennreal_numeral[symmetric]
```
```   292                        ennreal_plus[symmetric] ennreal_mult[symmetric] field_simps
```
```   293                     simp del: ennreal_numeral ennreal_plus)
```
```   294       finally show False
```
```   295         by simp
```
```   296     qed
```
```   297     then have "nn_integral M f < nn_integral M ?f'"
```
```   298       using \<open>0 < b\<close> \<open>nn_integral M f \<noteq> top\<close>
```
```   299       by (simp add: nn_integral_add nn_integral_cmult_indicator ennreal_zero_less_mult_iff zero_less_iff_neq_zero)
```
```   300     moreover
```
```   301     have "?f' \<in> G"
```
```   302       unfolding G_def
```
```   303     proof safe
```
```   304       fix X assume [measurable]: "X \<in> sets M"
```
```   305       have "(\<integral>\<^sup>+ x. ?f' x * indicator X x \<partial>M) = density M f (X - Y) + density M ?f (X \<inter> Y)"
```
```   306         by (auto simp add: emeasure_density nn_integral_add[symmetric] split: split_indicator intro!: nn_integral_cong)
```
```   307       also have "\<dots> \<le> N (X - Y) + N (X \<inter> Y)"
```
```   308         using G_D[OF \<open>f \<in> G\<close>] by (intro add_mono Y1) (auto simp: emeasure_density)
```
```   309       also have "\<dots> = N X"
```
```   310         by (subst plus_emeasure) (auto intro!: arg_cong2[where f=emeasure])
```
```   311       finally show "(\<integral>\<^sup>+ x. ?f' x * indicator X x \<partial>M) \<le> N X" .
```
```   312     qed simp
```
```   313     then have "nn_integral M ?f' \<le> ?y"
```
```   314       by (rule SUP_upper)
```
```   315     ultimately show False
```
```   316       by (simp add: int_f_eq_y)
```
```   317   qed
```
```   318   show ?thesis
```
```   319   proof (intro bexI[of _ f] measure_eqI conjI antisym)
```
```   320     fix A assume "A \<in> sets (density M f)" then show "emeasure (density M f) A \<le> emeasure N A"
```
```   321       by (auto simp: emeasure_density intro!: G_D[OF \<open>f \<in> G\<close>])
```
```   322   next
```
```   323     fix A assume A: "A \<in> sets (density M f)" then show "emeasure N A \<le> emeasure (density M f) A"
```
```   324       using upper_bound by auto
```
```   325   qed auto
```
```   326 qed
```
```   327
```
```   328 lemma (in finite_measure) split_space_into_finite_sets_and_rest:
```
```   329   assumes ac: "absolutely_continuous M N" and sets_eq[simp]: "sets N = sets M"
```
```   330   shows "\<exists>B::nat\<Rightarrow>'a set. disjoint_family B \<and> range B \<subseteq> sets M \<and> (\<forall>i. N (B i) \<noteq> \<infinity>) \<and>
```
```   331     (\<forall>A\<in>sets M. A \<inter> (\<Union>i. B i) = {} \<longrightarrow> (emeasure M A = 0 \<and> N A = 0) \<or> (emeasure M A > 0 \<and> N A = \<infinity>))"
```
```   332 proof -
```
```   333   let ?Q = "{Q\<in>sets M. N Q \<noteq> \<infinity>}"
```
```   334   let ?a = "SUP Q:?Q. emeasure M Q"
```
```   335   have "{} \<in> ?Q" by auto
```
```   336   then have Q_not_empty: "?Q \<noteq> {}" by blast
```
```   337   have "?a \<le> emeasure M (space M)" using sets.sets_into_space
```
```   338     by (auto intro!: SUP_least emeasure_mono)
```
```   339   then have "?a \<noteq> \<infinity>"
```
```   340     using finite_emeasure_space
```
```   341     by (auto simp: less_top[symmetric] top_unique simp del: SUP_eq_top_iff Sup_eq_top_iff)
```
```   342   from ennreal_SUP_countable_SUP [OF Q_not_empty, of "emeasure M"]
```
```   343   obtain Q'' where "range Q'' \<subseteq> emeasure M ` ?Q" and a: "?a = (SUP i::nat. Q'' i)"
```
```   344     by auto
```
```   345   then have "\<forall>i. \<exists>Q'. Q'' i = emeasure M Q' \<and> Q' \<in> ?Q" by auto
```
```   346   from choice[OF this] obtain Q' where Q': "\<And>i. Q'' i = emeasure M (Q' i)" "\<And>i. Q' i \<in> ?Q"
```
```   347     by auto
```
```   348   then have a_Lim: "?a = (SUP i::nat. emeasure M (Q' i))" using a by simp
```
```   349   let ?O = "\<lambda>n. \<Union>i\<le>n. Q' i"
```
```   350   have Union: "(SUP i. emeasure M (?O i)) = emeasure M (\<Union>i. ?O i)"
```
```   351   proof (rule SUP_emeasure_incseq[of ?O])
```
```   352     show "range ?O \<subseteq> sets M" using Q' by auto
```
```   353     show "incseq ?O" by (fastforce intro!: incseq_SucI)
```
```   354   qed
```
```   355   have Q'_sets[measurable]: "\<And>i. Q' i \<in> sets M" using Q' by auto
```
```   356   have O_sets: "\<And>i. ?O i \<in> sets M" using Q' by auto
```
```   357   then have O_in_G: "\<And>i. ?O i \<in> ?Q"
```
```   358   proof (safe del: notI)
```
```   359     fix i have "Q' ` {..i} \<subseteq> sets M" using Q' by auto
```
```   360     then have "N (?O i) \<le> (\<Sum>i\<le>i. N (Q' i))"
```
```   361       by (simp add: emeasure_subadditive_finite)
```
```   362     also have "\<dots> < \<infinity>" using Q' by (simp add: less_top)
```
```   363     finally show "N (?O i) \<noteq> \<infinity>" by simp
```
```   364   qed auto
```
```   365   have O_mono: "\<And>n. ?O n \<subseteq> ?O (Suc n)" by fastforce
```
```   366   have a_eq: "?a = emeasure M (\<Union>i. ?O i)" unfolding Union[symmetric]
```
```   367   proof (rule antisym)
```
```   368     show "?a \<le> (SUP i. emeasure M (?O i))" unfolding a_Lim
```
```   369       using Q' by (auto intro!: SUP_mono emeasure_mono)
```
```   370     show "(SUP i. emeasure M (?O i)) \<le> ?a"
```
```   371     proof (safe intro!: Sup_mono, unfold bex_simps)
```
```   372       fix i
```
```   373       have *: "(\<Union>(Q' ` {..i})) = ?O i" by auto
```
```   374       then show "\<exists>x. (x \<in> sets M \<and> N x \<noteq> \<infinity>) \<and>
```
```   375         emeasure M (\<Union>(Q' ` {..i})) \<le> emeasure M x"
```
```   376         using O_in_G[of i] by (auto intro!: exI[of _ "?O i"])
```
```   377     qed
```
```   378   qed
```
```   379   let ?O_0 = "(\<Union>i. ?O i)"
```
```   380   have "?O_0 \<in> sets M" using Q' by auto
```
```   381   have "disjointed Q' i \<in> sets M" for i
```
```   382     using sets.range_disjointed_sets[of Q' M] using Q'_sets by (auto simp: subset_eq)
```
```   383   note Q_sets = this
```
```   384   show ?thesis
```
```   385   proof (intro bexI exI conjI ballI impI allI)
```
```   386     show "disjoint_family (disjointed Q')"
```
```   387       by (rule disjoint_family_disjointed)
```
```   388     show "range (disjointed Q') \<subseteq> sets M"
```
```   389       using Q'_sets by (intro sets.range_disjointed_sets) auto
```
```   390     { fix A assume A: "A \<in> sets M" "A \<inter> (\<Union>i. disjointed Q' i) = {}"
```
```   391       then have A1: "A \<inter> (\<Union>i. Q' i) = {}"
```
```   392         unfolding UN_disjointed_eq by auto
```
```   393       show "emeasure M A = 0 \<and> N A = 0 \<or> 0 < emeasure M A \<and> N A = \<infinity>"
```
```   394       proof (rule disjCI, simp)
```
```   395         assume *: "emeasure M A = 0 \<or> N A \<noteq> top"
```
```   396         show "emeasure M A = 0 \<and> N A = 0"
```
```   397         proof (cases "emeasure M A = 0")
```
```   398           case True
```
```   399           with ac A have "N A = 0"
```
```   400             unfolding absolutely_continuous_def by auto
```
```   401           with True show ?thesis by simp
```
```   402         next
```
```   403           case False
```
```   404           with * have "N A \<noteq> \<infinity>" by auto
```
```   405           with A have "emeasure M ?O_0 + emeasure M A = emeasure M (?O_0 \<union> A)"
```
```   406             using Q' A1 by (auto intro!: plus_emeasure simp: set_eq_iff)
```
```   407           also have "\<dots> = (SUP i. emeasure M (?O i \<union> A))"
```
```   408           proof (rule SUP_emeasure_incseq[of "\<lambda>i. ?O i \<union> A", symmetric, simplified])
```
```   409             show "range (\<lambda>i. ?O i \<union> A) \<subseteq> sets M"
```
```   410               using \<open>N A \<noteq> \<infinity>\<close> O_sets A by auto
```
```   411           qed (fastforce intro!: incseq_SucI)
```
```   412           also have "\<dots> \<le> ?a"
```
```   413           proof (safe intro!: SUP_least)
```
```   414             fix i have "?O i \<union> A \<in> ?Q"
```
```   415             proof (safe del: notI)
```
```   416               show "?O i \<union> A \<in> sets M" using O_sets A by auto
```
```   417               from O_in_G[of i] have "N (?O i \<union> A) \<le> N (?O i) + N A"
```
```   418                 using emeasure_subadditive[of "?O i" N A] A O_sets by auto
```
```   419               with O_in_G[of i] show "N (?O i \<union> A) \<noteq> \<infinity>"
```
```   420                 using \<open>N A \<noteq> \<infinity>\<close> by (auto simp: top_unique)
```
```   421             qed
```
```   422             then show "emeasure M (?O i \<union> A) \<le> ?a" by (rule SUP_upper)
```
```   423           qed
```
```   424           finally have "emeasure M A = 0"
```
```   425             unfolding a_eq using measure_nonneg[of M A] by (simp add: emeasure_eq_measure)
```
```   426           with \<open>emeasure M A \<noteq> 0\<close> show ?thesis by auto
```
```   427         qed
```
```   428       qed }
```
```   429     { fix i
```
```   430       have "N (disjointed Q' i) \<le> N (Q' i)"
```
```   431         by (auto intro!: emeasure_mono simp: disjointed_def)
```
```   432       then show "N (disjointed Q' i) \<noteq> \<infinity>"
```
```   433         using Q'(2)[of i] by (auto simp: top_unique) }
```
```   434   qed
```
```   435 qed
```
```   436
```
```   437 lemma (in finite_measure) Radon_Nikodym_finite_measure_infinite:
```
```   438   assumes "absolutely_continuous M N" and sets_eq: "sets N = sets M"
```
```   439   shows "\<exists>f\<in>borel_measurable M. density M f = N"
```
```   440 proof -
```
```   441   from split_space_into_finite_sets_and_rest[OF assms]
```
```   442   obtain Q :: "nat \<Rightarrow> 'a set"
```
```   443     where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
```
```   444     and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<inter> (\<Union>i. Q i) = {} \<Longrightarrow> emeasure M A = 0 \<and> N A = 0 \<or> 0 < emeasure M A \<and> N A = \<infinity>"
```
```   445     and Q_fin: "\<And>i. N (Q i) \<noteq> \<infinity>" by force
```
```   446   from Q have Q_sets: "\<And>i. Q i \<in> sets M" by auto
```
```   447   let ?N = "\<lambda>i. density N (indicator (Q i))" and ?M = "\<lambda>i. density M (indicator (Q i))"
```
```   448   have "\<forall>i. \<exists>f\<in>borel_measurable (?M i). density (?M i) f = ?N i"
```
```   449   proof (intro allI finite_measure.Radon_Nikodym_finite_measure)
```
```   450     fix i
```
```   451     from Q show "finite_measure (?M i)"
```
```   452       by (auto intro!: finite_measureI cong: nn_integral_cong
```
```   453                simp add: emeasure_density subset_eq sets_eq)
```
```   454     from Q have "emeasure (?N i) (space N) = emeasure N (Q i)"
```
```   455       by (simp add: sets_eq[symmetric] emeasure_density subset_eq cong: nn_integral_cong)
```
```   456     with Q_fin show "finite_measure (?N i)"
```
```   457       by (auto intro!: finite_measureI)
```
```   458     show "sets (?N i) = sets (?M i)" by (simp add: sets_eq)
```
```   459     have [measurable]: "\<And>A. A \<in> sets M \<Longrightarrow> A \<in> sets N" by (simp add: sets_eq)
```
```   460     show "absolutely_continuous (?M i) (?N i)"
```
```   461       using \<open>absolutely_continuous M N\<close> \<open>Q i \<in> sets M\<close>
```
```   462       by (auto simp: absolutely_continuous_def null_sets_density_iff sets_eq
```
```   463                intro!: absolutely_continuous_AE[OF sets_eq])
```
```   464   qed
```
```   465   from choice[OF this[unfolded Bex_def]]
```
```   466   obtain f where borel: "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
```
```   467     and f_density: "\<And>i. density (?M i) (f i) = ?N i"
```
```   468     by force
```
```   469   { fix A i assume A: "A \<in> sets M"
```
```   470     with Q borel have "(\<integral>\<^sup>+x. f i x * indicator (Q i \<inter> A) x \<partial>M) = emeasure (density (?M i) (f i)) A"
```
```   471       by (auto simp add: emeasure_density nn_integral_density subset_eq
```
```   472                intro!: nn_integral_cong split: split_indicator)
```
```   473     also have "\<dots> = emeasure N (Q i \<inter> A)"
```
```   474       using A Q by (simp add: f_density emeasure_restricted subset_eq sets_eq)
```
```   475     finally have "emeasure N (Q i \<inter> A) = (\<integral>\<^sup>+x. f i x * indicator (Q i \<inter> A) x \<partial>M)" .. }
```
```   476   note integral_eq = this
```
```   477   let ?f = "\<lambda>x. (\<Sum>i. f i x * indicator (Q i) x) + \<infinity> * indicator (space M - (\<Union>i. Q i)) x"
```
```   478   show ?thesis
```
```   479   proof (safe intro!: bexI[of _ ?f])
```
```   480     show "?f \<in> borel_measurable M" using borel Q_sets
```
```   481       by (auto intro!: measurable_If)
```
```   482     show "density M ?f = N"
```
```   483     proof (rule measure_eqI)
```
```   484       fix A assume "A \<in> sets (density M ?f)"
```
```   485       then have "A \<in> sets M" by simp
```
```   486       have Qi: "\<And>i. Q i \<in> sets M" using Q by auto
```
```   487       have [intro,simp]: "\<And>i. (\<lambda>x. f i x * indicator (Q i \<inter> A) x) \<in> borel_measurable M"
```
```   488         "\<And>i. AE x in M. 0 \<le> f i x * indicator (Q i \<inter> A) x"
```
```   489         using borel Qi \<open>A \<in> sets M\<close> by auto
```
```   490       have "(\<integral>\<^sup>+x. ?f x * indicator A x \<partial>M) = (\<integral>\<^sup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) + \<infinity> * indicator ((space M - (\<Union>i. Q i)) \<inter> A) x \<partial>M)"
```
```   491         using borel by (intro nn_integral_cong) (auto simp: indicator_def)
```
```   492       also have "\<dots> = (\<integral>\<^sup>+x. (\<Sum>i. f i x * indicator (Q i \<inter> A) x) \<partial>M) + \<infinity> * emeasure M ((space M - (\<Union>i. Q i)) \<inter> A)"
```
```   493         using borel Qi \<open>A \<in> sets M\<close>
```
```   494         by (subst nn_integral_add)
```
```   495            (auto simp add: nn_integral_cmult_indicator sets.Int intro!: suminf_0_le)
```
```   496       also have "\<dots> = (\<Sum>i. N (Q i \<inter> A)) + \<infinity> * emeasure M ((space M - (\<Union>i. Q i)) \<inter> A)"
```
```   497         by (subst integral_eq[OF \<open>A \<in> sets M\<close>], subst nn_integral_suminf) auto
```
```   498       finally have "(\<integral>\<^sup>+x. ?f x * indicator A x \<partial>M) = (\<Sum>i. N (Q i \<inter> A)) + \<infinity> * emeasure M ((space M - (\<Union>i. Q i)) \<inter> A)" .
```
```   499       moreover have "(\<Sum>i. N (Q i \<inter> A)) = N ((\<Union>i. Q i) \<inter> A)"
```
```   500         using Q Q_sets \<open>A \<in> sets M\<close>
```
```   501         by (subst suminf_emeasure) (auto simp: disjoint_family_on_def sets_eq)
```
```   502       moreover
```
```   503       have "(space M - (\<Union>x. Q x)) \<inter> A \<inter> (\<Union>x. Q x) = {}"
```
```   504         by auto
```
```   505       then have "\<infinity> * emeasure M ((space M - (\<Union>i. Q i)) \<inter> A) = N ((space M - (\<Union>i. Q i)) \<inter> A)"
```
```   506         using in_Q0[of "(space M - (\<Union>i. Q i)) \<inter> A"] \<open>A \<in> sets M\<close> Q by (auto simp: ennreal_top_mult)
```
```   507       moreover have "(space M - (\<Union>i. Q i)) \<inter> A \<in> sets M" "((\<Union>i. Q i) \<inter> A) \<in> sets M"
```
```   508         using Q_sets \<open>A \<in> sets M\<close> by auto
```
```   509       moreover have "((\<Union>i. Q i) \<inter> A) \<union> ((space M - (\<Union>i. Q i)) \<inter> A) = A" "((\<Union>i. Q i) \<inter> A) \<inter> ((space M - (\<Union>i. Q i)) \<inter> A) = {}"
```
```   510         using \<open>A \<in> sets M\<close> sets.sets_into_space by auto
```
```   511       ultimately have "N A = (\<integral>\<^sup>+x. ?f x * indicator A x \<partial>M)"
```
```   512         using plus_emeasure[of "(\<Union>i. Q i) \<inter> A" N "(space M - (\<Union>i. Q i)) \<inter> A"] by (simp add: sets_eq)
```
```   513       with \<open>A \<in> sets M\<close> borel Q show "emeasure (density M ?f) A = N A"
```
```   514         by (auto simp: subset_eq emeasure_density)
```
```   515     qed (simp add: sets_eq)
```
```   516   qed
```
```   517 qed
```
```   518
```
```   519 lemma (in sigma_finite_measure) Radon_Nikodym:
```
```   520   assumes ac: "absolutely_continuous M N" assumes sets_eq: "sets N = sets M"
```
```   521   shows "\<exists>f \<in> borel_measurable M. density M f = N"
```
```   522 proof -
```
```   523   from Ex_finite_integrable_function
```
```   524   obtain h where finite: "integral\<^sup>N M h \<noteq> \<infinity>" and
```
```   525     borel: "h \<in> borel_measurable M" and
```
```   526     nn: "\<And>x. 0 \<le> h x" and
```
```   527     pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x" and
```
```   528     "\<And>x. x \<in> space M \<Longrightarrow> h x < \<infinity>" by auto
```
```   529   let ?T = "\<lambda>A. (\<integral>\<^sup>+x. h x * indicator A x \<partial>M)"
```
```   530   let ?MT = "density M h"
```
```   531   from borel finite nn interpret T: finite_measure ?MT
```
```   532     by (auto intro!: finite_measureI cong: nn_integral_cong simp: emeasure_density)
```
```   533   have "absolutely_continuous ?MT N" "sets N = sets ?MT"
```
```   534   proof (unfold absolutely_continuous_def, safe)
```
```   535     fix A assume "A \<in> null_sets ?MT"
```
```   536     with borel have "A \<in> sets M" "AE x in M. x \<in> A \<longrightarrow> h x \<le> 0"
```
```   537       by (auto simp add: null_sets_density_iff)
```
```   538     with pos sets.sets_into_space have "AE x in M. x \<notin> A"
```
```   539       by (elim eventually_mono) (auto simp: not_le[symmetric])
```
```   540     then have "A \<in> null_sets M"
```
```   541       using \<open>A \<in> sets M\<close> by (simp add: AE_iff_null_sets)
```
```   542     with ac show "A \<in> null_sets N"
```
```   543       by (auto simp: absolutely_continuous_def)
```
```   544   qed (auto simp add: sets_eq)
```
```   545   from T.Radon_Nikodym_finite_measure_infinite[OF this]
```
```   546   obtain f where f_borel: "f \<in> borel_measurable M" "density ?MT f = N" by auto
```
```   547   with nn borel show ?thesis
```
```   548     by (auto intro!: bexI[of _ "\<lambda>x. h x * f x"] simp: density_density_eq)
```
```   549 qed
```
```   550
```
```   551 subsection \<open>Uniqueness of densities\<close>
```
```   552
```
```   553 lemma finite_density_unique:
```
```   554   assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```   555   assumes pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> g x"
```
```   556   and fin: "integral\<^sup>N M f \<noteq> \<infinity>"
```
```   557   shows "density M f = density M g \<longleftrightarrow> (AE x in M. f x = g x)"
```
```   558 proof (intro iffI ballI)
```
```   559   fix A assume eq: "AE x in M. f x = g x"
```
```   560   with borel show "density M f = density M g"
```
```   561     by (auto intro: density_cong)
```
```   562 next
```
```   563   let ?P = "\<lambda>f A. \<integral>\<^sup>+ x. f x * indicator A x \<partial>M"
```
```   564   assume "density M f = density M g"
```
```   565   with borel have eq: "\<forall>A\<in>sets M. ?P f A = ?P g A"
```
```   566     by (simp add: emeasure_density[symmetric])
```
```   567   from this[THEN bspec, OF sets.top] fin
```
```   568   have g_fin: "integral\<^sup>N M g \<noteq> \<infinity>" by (simp cong: nn_integral_cong)
```
```   569   { fix f g assume borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```   570       and pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> g x"
```
```   571       and g_fin: "integral\<^sup>N M g \<noteq> \<infinity>" and eq: "\<forall>A\<in>sets M. ?P f A = ?P g A"
```
```   572     let ?N = "{x\<in>space M. g x < f x}"
```
```   573     have N: "?N \<in> sets M" using borel by simp
```
```   574     have "?P g ?N \<le> integral\<^sup>N M g" using pos
```
```   575       by (intro nn_integral_mono_AE) (auto split: split_indicator)
```
```   576     then have Pg_fin: "?P g ?N \<noteq> \<infinity>" using g_fin by (auto simp: top_unique)
```
```   577     have "?P (\<lambda>x. (f x - g x)) ?N = (\<integral>\<^sup>+x. f x * indicator ?N x - g x * indicator ?N x \<partial>M)"
```
```   578       by (auto intro!: nn_integral_cong simp: indicator_def)
```
```   579     also have "\<dots> = ?P f ?N - ?P g ?N"
```
```   580     proof (rule nn_integral_diff)
```
```   581       show "(\<lambda>x. f x * indicator ?N x) \<in> borel_measurable M" "(\<lambda>x. g x * indicator ?N x) \<in> borel_measurable M"
```
```   582         using borel N by auto
```
```   583       show "AE x in M. g x * indicator ?N x \<le> f x * indicator ?N x"
```
```   584         using pos by (auto split: split_indicator)
```
```   585     qed fact
```
```   586     also have "\<dots> = 0"
```
```   587       unfolding eq[THEN bspec, OF N] using Pg_fin by auto
```
```   588     finally have "AE x in M. f x \<le> g x"
```
```   589       using pos borel nn_integral_PInf_AE[OF borel(2) g_fin]
```
```   590       by (subst (asm) nn_integral_0_iff_AE)
```
```   591          (auto split: split_indicator simp: not_less ennreal_minus_eq_0) }
```
```   592   from this[OF borel pos g_fin eq] this[OF borel(2,1) pos(2,1) fin] eq
```
```   593   show "AE x in M. f x = g x" by auto
```
```   594 qed
```
```   595
```
```   596 lemma (in finite_measure) density_unique_finite_measure:
```
```   597   assumes borel: "f \<in> borel_measurable M" "f' \<in> borel_measurable M"
```
```   598   assumes pos: "AE x in M. 0 \<le> f x" "AE x in M. 0 \<le> f' x"
```
```   599   assumes f: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^sup>+x. f' x * indicator A x \<partial>M)"
```
```   600     (is "\<And>A. A \<in> sets M \<Longrightarrow> ?P f A = ?P f' A")
```
```   601   shows "AE x in M. f x = f' x"
```
```   602 proof -
```
```   603   let ?D = "\<lambda>f. density M f"
```
```   604   let ?N = "\<lambda>A. ?P f A" and ?N' = "\<lambda>A. ?P f' A"
```
```   605   let ?f = "\<lambda>A x. f x * indicator A x" and ?f' = "\<lambda>A x. f' x * indicator A x"
```
```   606
```
```   607   have ac: "absolutely_continuous M (density M f)" "sets (density M f) = sets M"
```
```   608     using borel by (auto intro!: absolutely_continuousI_density)
```
```   609   from split_space_into_finite_sets_and_rest[OF this]
```
```   610   obtain Q :: "nat \<Rightarrow> 'a set"
```
```   611     where Q: "disjoint_family Q" "range Q \<subseteq> sets M"
```
```   612     and in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<inter> (\<Union>i. Q i) = {} \<Longrightarrow> emeasure M A = 0 \<and> ?D f A = 0 \<or> 0 < emeasure M A \<and> ?D f A = \<infinity>"
```
```   613     and Q_fin: "\<And>i. ?D f (Q i) \<noteq> \<infinity>" by force
```
```   614   with borel pos have in_Q0: "\<And>A. A \<in> sets M \<Longrightarrow> A \<inter> (\<Union>i. Q i) = {} \<Longrightarrow> emeasure M A = 0 \<and> ?N A = 0 \<or> 0 < emeasure M A \<and> ?N A = \<infinity>"
```
```   615     and Q_fin: "\<And>i. ?N (Q i) \<noteq> \<infinity>" by (auto simp: emeasure_density subset_eq)
```
```   616
```
```   617   from Q have Q_sets[measurable]: "\<And>i. Q i \<in> sets M" by auto
```
```   618   let ?D = "{x\<in>space M. f x \<noteq> f' x}"
```
```   619   have "?D \<in> sets M" using borel by auto
```
```   620   have *: "\<And>i x A. \<And>y::ennreal. y * indicator (Q i) x * indicator A x = y * indicator (Q i \<inter> A) x"
```
```   621     unfolding indicator_def by auto
```
```   622   have "\<forall>i. AE x in M. ?f (Q i) x = ?f' (Q i) x" using borel Q_fin Q pos
```
```   623     by (intro finite_density_unique[THEN iffD1] allI)
```
```   624        (auto intro!: f measure_eqI simp: emeasure_density * subset_eq)
```
```   625   moreover have "AE x in M. ?f (space M - (\<Union>i. Q i)) x = ?f' (space M - (\<Union>i. Q i)) x"
```
```   626   proof (rule AE_I')
```
```   627     { fix f :: "'a \<Rightarrow> ennreal" assume borel: "f \<in> borel_measurable M"
```
```   628         and eq: "\<And>A. A \<in> sets M \<Longrightarrow> ?N A = (\<integral>\<^sup>+x. f x * indicator A x \<partial>M)"
```
```   629       let ?A = "\<lambda>i. (space M - (\<Union>i. Q i)) \<inter> {x \<in> space M. f x < (i::nat)}"
```
```   630       have "(\<Union>i. ?A i) \<in> null_sets M"
```
```   631       proof (rule null_sets_UN)
```
```   632         fix i ::nat have "?A i \<in> sets M"
```
```   633           using borel by auto
```
```   634         have "?N (?A i) \<le> (\<integral>\<^sup>+x. (i::ennreal) * indicator (?A i) x \<partial>M)"
```
```   635           unfolding eq[OF \<open>?A i \<in> sets M\<close>]
```
```   636           by (auto intro!: nn_integral_mono simp: indicator_def)
```
```   637         also have "\<dots> = i * emeasure M (?A i)"
```
```   638           using \<open>?A i \<in> sets M\<close> by (auto intro!: nn_integral_cmult_indicator)
```
```   639         also have "\<dots> < \<infinity>" using emeasure_real[of "?A i"] by (auto simp: ennreal_mult_less_top of_nat_less_top)
```
```   640         finally have "?N (?A i) \<noteq> \<infinity>" by simp
```
```   641         then show "?A i \<in> null_sets M" using in_Q0[OF \<open>?A i \<in> sets M\<close>] \<open>?A i \<in> sets M\<close> by auto
```
```   642       qed
```
```   643       also have "(\<Union>i. ?A i) = (space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f x \<noteq> \<infinity>}"
```
```   644         by (auto simp: ennreal_Ex_less_of_nat less_top[symmetric])
```
```   645       finally have "(space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets M" by simp }
```
```   646     from this[OF borel(1) refl] this[OF borel(2) f]
```
```   647     have "(space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f x \<noteq> \<infinity>} \<in> null_sets M" "(space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f' x \<noteq> \<infinity>} \<in> null_sets M" by simp_all
```
```   648     then show "((space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f x \<noteq> \<infinity>}) \<union> ((space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f' x \<noteq> \<infinity>}) \<in> null_sets M" by (rule null_sets.Un)
```
```   649     show "{x \<in> space M. ?f (space M - (\<Union>i. Q i)) x \<noteq> ?f' (space M - (\<Union>i. Q i)) x} \<subseteq>
```
```   650       ((space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f x \<noteq> \<infinity>}) \<union> ((space M - (\<Union>i. Q i)) \<inter> {x\<in>space M. f' x \<noteq> \<infinity>})" by (auto simp: indicator_def)
```
```   651   qed
```
```   652   moreover have "AE x in M. (?f (space M - (\<Union>i. Q i)) x = ?f' (space M - (\<Union>i. Q i)) x) \<longrightarrow> (\<forall>i. ?f (Q i) x = ?f' (Q i) x) \<longrightarrow>
```
```   653     ?f (space M) x = ?f' (space M) x"
```
```   654     by (auto simp: indicator_def)
```
```   655   ultimately have "AE x in M. ?f (space M) x = ?f' (space M) x"
```
```   656     unfolding AE_all_countable[symmetric]
```
```   657     by eventually_elim (auto split: if_split_asm simp: indicator_def)
```
```   658   then show "AE x in M. f x = f' x" by auto
```
```   659 qed
```
```   660
```
```   661 lemma (in sigma_finite_measure) density_unique:
```
```   662   assumes f: "f \<in> borel_measurable M"
```
```   663   assumes f': "f' \<in> borel_measurable M"
```
```   664   assumes density_eq: "density M f = density M f'"
```
```   665   shows "AE x in M. f x = f' x"
```
```   666 proof -
```
```   667   obtain h where h_borel: "h \<in> borel_measurable M"
```
```   668     and fin: "integral\<^sup>N M h \<noteq> \<infinity>" and pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 < h x \<and> h x < \<infinity>" "\<And>x. 0 \<le> h x"
```
```   669     using Ex_finite_integrable_function by auto
```
```   670   then have h_nn: "AE x in M. 0 \<le> h x" by auto
```
```   671   let ?H = "density M h"
```
```   672   interpret h: finite_measure ?H
```
```   673     using fin h_borel pos
```
```   674     by (intro finite_measureI) (simp cong: nn_integral_cong emeasure_density add: fin)
```
```   675   let ?fM = "density M f"
```
```   676   let ?f'M = "density M f'"
```
```   677   { fix A assume "A \<in> sets M"
```
```   678     then have "{x \<in> space M. h x * indicator A x \<noteq> 0} = A"
```
```   679       using pos(1) sets.sets_into_space by (force simp: indicator_def)
```
```   680     then have "(\<integral>\<^sup>+x. h x * indicator A x \<partial>M) = 0 \<longleftrightarrow> A \<in> null_sets M"
```
```   681       using h_borel \<open>A \<in> sets M\<close> h_nn by (subst nn_integral_0_iff) auto }
```
```   682   note h_null_sets = this
```
```   683   { fix A assume "A \<in> sets M"
```
```   684     have "(\<integral>\<^sup>+x. f x * (h x * indicator A x) \<partial>M) = (\<integral>\<^sup>+x. h x * indicator A x \<partial>?fM)"
```
```   685       using \<open>A \<in> sets M\<close> h_borel h_nn f f'
```
```   686       by (intro nn_integral_density[symmetric]) auto
```
```   687     also have "\<dots> = (\<integral>\<^sup>+x. h x * indicator A x \<partial>?f'M)"
```
```   688       by (simp_all add: density_eq)
```
```   689     also have "\<dots> = (\<integral>\<^sup>+x. f' x * (h x * indicator A x) \<partial>M)"
```
```   690       using \<open>A \<in> sets M\<close> h_borel h_nn f f'
```
```   691       by (intro nn_integral_density) auto
```
```   692     finally have "(\<integral>\<^sup>+x. h x * (f x * indicator A x) \<partial>M) = (\<integral>\<^sup>+x. h x * (f' x * indicator A x) \<partial>M)"
```
```   693       by (simp add: ac_simps)
```
```   694     then have "(\<integral>\<^sup>+x. (f x * indicator A x) \<partial>?H) = (\<integral>\<^sup>+x. (f' x * indicator A x) \<partial>?H)"
```
```   695       using \<open>A \<in> sets M\<close> h_borel h_nn f f'
```
```   696       by (subst (asm) (1 2) nn_integral_density[symmetric]) auto }
```
```   697   then have "AE x in ?H. f x = f' x" using h_borel h_nn f f'
```
```   698     by (intro h.density_unique_finite_measure absolutely_continuous_AE[of M]) auto
```
```   699   with AE_space[of M] pos show "AE x in M. f x = f' x"
```
```   700     unfolding AE_density[OF h_borel] by auto
```
```   701 qed
```
```   702
```
```   703 lemma (in sigma_finite_measure) density_unique_iff:
```
```   704   assumes f: "f \<in> borel_measurable M" and f': "f' \<in> borel_measurable M"
```
```   705   shows "density M f = density M f' \<longleftrightarrow> (AE x in M. f x = f' x)"
```
```   706   using density_unique[OF assms] density_cong[OF f f'] by auto
```
```   707
```
```   708 lemma sigma_finite_density_unique:
```
```   709   assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
```
```   710   and fin: "sigma_finite_measure (density M f)"
```
```   711   shows "density M f = density M g \<longleftrightarrow> (AE x in M. f x = g x)"
```
```   712 proof
```
```   713   assume "AE x in M. f x = g x" with borel show "density M f = density M g"
```
```   714     by (auto intro: density_cong)
```
```   715 next
```
```   716   assume eq: "density M f = density M g"
```
```   717   interpret f: sigma_finite_measure "density M f" by fact
```
```   718   from f.sigma_finite_incseq guess A . note cover = this
```
```   719
```
```   720   have "AE x in M. \<forall>i. x \<in> A i \<longrightarrow> f x = g x"
```
```   721     unfolding AE_all_countable
```
```   722   proof
```
```   723     fix i
```
```   724     have "density (density M f) (indicator (A i)) = density (density M g) (indicator (A i))"
```
```   725       unfolding eq ..
```
```   726     moreover have "(\<integral>\<^sup>+x. f x * indicator (A i) x \<partial>M) \<noteq> \<infinity>"
```
```   727       using cover(1) cover(3)[of i] borel by (auto simp: emeasure_density subset_eq)
```
```   728     ultimately have "AE x in M. f x * indicator (A i) x = g x * indicator (A i) x"
```
```   729       using borel cover(1)
```
```   730       by (intro finite_density_unique[THEN iffD1]) (auto simp: density_density_eq subset_eq)
```
```   731     then show "AE x in M. x \<in> A i \<longrightarrow> f x = g x"
```
```   732       by auto
```
```   733   qed
```
```   734   with AE_space show "AE x in M. f x = g x"
```
```   735     apply eventually_elim
```
```   736     using cover(2)[symmetric]
```
```   737     apply auto
```
```   738     done
```
```   739 qed
```
```   740
```
```   741 lemma (in sigma_finite_measure) sigma_finite_iff_density_finite':
```
```   742   assumes f: "f \<in> borel_measurable M"
```
```   743   shows "sigma_finite_measure (density M f) \<longleftrightarrow> (AE x in M. f x \<noteq> \<infinity>)"
```
```   744     (is "sigma_finite_measure ?N \<longleftrightarrow> _")
```
```   745 proof
```
```   746   assume "sigma_finite_measure ?N"
```
```   747   then interpret N: sigma_finite_measure ?N .
```
```   748   from N.Ex_finite_integrable_function obtain h where
```
```   749     h: "h \<in> borel_measurable M" "integral\<^sup>N ?N h \<noteq> \<infinity>" and
```
```   750     fin: "\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>"
```
```   751     by auto
```
```   752   have "AE x in M. f x * h x \<noteq> \<infinity>"
```
```   753   proof (rule AE_I')
```
```   754     have "integral\<^sup>N ?N h = (\<integral>\<^sup>+x. f x * h x \<partial>M)"
```
```   755       using f h by (auto intro!: nn_integral_density)
```
```   756     then have "(\<integral>\<^sup>+x. f x * h x \<partial>M) \<noteq> \<infinity>"
```
```   757       using h(2) by simp
```
```   758     then show "(\<lambda>x. f x * h x) -` {\<infinity>} \<inter> space M \<in> null_sets M"
```
```   759       using f h(1) by (auto intro!: nn_integral_PInf[unfolded infinity_ennreal_def] borel_measurable_vimage)
```
```   760   qed auto
```
```   761   then show "AE x in M. f x \<noteq> \<infinity>"
```
```   762     using fin by (auto elim!: AE_Ball_mp simp: less_top ennreal_mult_less_top)
```
```   763 next
```
```   764   assume AE: "AE x in M. f x \<noteq> \<infinity>"
```
```   765   from sigma_finite guess Q . note Q = this
```
```   766   define A where "A i =
```
```   767     f -` (case i of 0 \<Rightarrow> {\<infinity>} | Suc n \<Rightarrow> {.. ennreal(of_nat (Suc n))}) \<inter> space M" for i
```
```   768   { fix i j have "A i \<inter> Q j \<in> sets M"
```
```   769     unfolding A_def using f Q
```
```   770     apply (rule_tac sets.Int)
```
```   771     by (cases i) (auto intro: measurable_sets[OF f(1)]) }
```
```   772   note A_in_sets = this
```
```   773
```
```   774   show "sigma_finite_measure ?N"
```
```   775   proof (standard, intro exI conjI ballI)
```
```   776     show "countable (range (\<lambda>(i, j). A i \<inter> Q j))"
```
```   777       by auto
```
```   778     show "range (\<lambda>(i, j). A i \<inter> Q j) \<subseteq> sets (density M f)"
```
```   779       using A_in_sets by auto
```
```   780   next
```
```   781     have "\<Union>range (\<lambda>(i, j). A i \<inter> Q j) = (\<Union>i j. A i \<inter> Q j)"
```
```   782       by auto
```
```   783     also have "\<dots> = (\<Union>i. A i) \<inter> space M" using Q by auto
```
```   784     also have "(\<Union>i. A i) = space M"
```
```   785     proof safe
```
```   786       fix x assume x: "x \<in> space M"
```
```   787       show "x \<in> (\<Union>i. A i)"
```
```   788       proof (cases "f x" rule: ennreal_cases)
```
```   789         case top with x show ?thesis unfolding A_def by (auto intro: exI[of _ 0])
```
```   790       next
```
```   791         case (real r)
```
```   792         with ennreal_Ex_less_of_nat[of "f x"] obtain n :: nat where "f x < n"
```
```   793           by auto
```
```   794         also have "n < (Suc n :: ennreal)"
```
```   795           by simp
```
```   796         finally show ?thesis
```
```   797           using x real by (auto simp: A_def ennreal_of_nat_eq_real_of_nat intro!: exI[of _ "Suc n"])
```
```   798       qed
```
```   799     qed (auto simp: A_def)
```
```   800     finally show "\<Union>range (\<lambda>(i, j). A i \<inter> Q j) = space ?N" by simp
```
```   801   next
```
```   802     fix X assume "X \<in> range (\<lambda>(i, j). A i \<inter> Q j)"
```
```   803     then obtain i j where [simp]:"X = A i \<inter> Q j" by auto
```
```   804     have "(\<integral>\<^sup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<noteq> \<infinity>"
```
```   805     proof (cases i)
```
```   806       case 0
```
```   807       have "AE x in M. f x * indicator (A i \<inter> Q j) x = 0"
```
```   808         using AE by (auto simp: A_def \<open>i = 0\<close>)
```
```   809       from nn_integral_cong_AE[OF this] show ?thesis by simp
```
```   810     next
```
```   811       case (Suc n)
```
```   812       then have "(\<integral>\<^sup>+x. f x * indicator (A i \<inter> Q j) x \<partial>M) \<le>
```
```   813         (\<integral>\<^sup>+x. (Suc n :: ennreal) * indicator (Q j) x \<partial>M)"
```
```   814         by (auto intro!: nn_integral_mono simp: indicator_def A_def ennreal_of_nat_eq_real_of_nat)
```
```   815       also have "\<dots> = Suc n * emeasure M (Q j)"
```
```   816         using Q by (auto intro!: nn_integral_cmult_indicator)
```
```   817       also have "\<dots> < \<infinity>"
```
```   818         using Q by (auto simp: ennreal_mult_less_top less_top of_nat_less_top)
```
```   819       finally show ?thesis by simp
```
```   820     qed
```
```   821     then show "emeasure ?N X \<noteq> \<infinity>"
```
```   822       using A_in_sets Q f by (auto simp: emeasure_density)
```
```   823   qed
```
```   824 qed
```
```   825
```
```   826 lemma (in sigma_finite_measure) sigma_finite_iff_density_finite:
```
```   827   "f \<in> borel_measurable M \<Longrightarrow> sigma_finite_measure (density M f) \<longleftrightarrow> (AE x in M. f x \<noteq> \<infinity>)"
```
```   828   by (subst sigma_finite_iff_density_finite')
```
```   829      (auto simp: max_def intro!: measurable_If)
```
```   830
```
```   831 subsection \<open>Radon-Nikodym derivative\<close>
```
```   832
```
```   833 definition RN_deriv :: "'a measure \<Rightarrow> 'a measure \<Rightarrow> 'a \<Rightarrow> ennreal" where
```
```   834   "RN_deriv M N =
```
```   835     (if \<exists>f. f \<in> borel_measurable M \<and> density M f = N
```
```   836        then SOME f. f \<in> borel_measurable M \<and> density M f = N
```
```   837        else (\<lambda>_. 0))"
```
```   838
```
```   839 lemma RN_derivI:
```
```   840   assumes "f \<in> borel_measurable M" "density M f = N"
```
```   841   shows "density M (RN_deriv M N) = N"
```
```   842 proof -
```
```   843   have *: "\<exists>f. f \<in> borel_measurable M \<and> density M f = N"
```
```   844     using assms by auto
```
```   845   then have "density M (SOME f. f \<in> borel_measurable M \<and> density M f = N) = N"
```
```   846     by (rule someI2_ex) auto
```
```   847   with * show ?thesis
```
```   848     by (auto simp: RN_deriv_def)
```
```   849 qed
```
```   850
```
```   851 lemma borel_measurable_RN_deriv[measurable]: "RN_deriv M N \<in> borel_measurable M"
```
```   852 proof -
```
```   853   { assume ex: "\<exists>f. f \<in> borel_measurable M \<and> density M f = N"
```
```   854     have 1: "(SOME f. f \<in> borel_measurable M \<and> density M f = N) \<in> borel_measurable M"
```
```   855       using ex by (rule someI2_ex) auto }
```
```   856   from this show ?thesis
```
```   857     by (auto simp: RN_deriv_def)
```
```   858 qed
```
```   859
```
```   860 lemma density_RN_deriv_density:
```
```   861   assumes f: "f \<in> borel_measurable M"
```
```   862   shows "density M (RN_deriv M (density M f)) = density M f"
```
```   863   by (rule RN_derivI[OF f]) simp
```
```   864
```
```   865 lemma (in sigma_finite_measure) density_RN_deriv:
```
```   866   "absolutely_continuous M N \<Longrightarrow> sets N = sets M \<Longrightarrow> density M (RN_deriv M N) = N"
```
```   867   by (metis RN_derivI Radon_Nikodym)
```
```   868
```
```   869 lemma (in sigma_finite_measure) RN_deriv_nn_integral:
```
```   870   assumes N: "absolutely_continuous M N" "sets N = sets M"
```
```   871     and f: "f \<in> borel_measurable M"
```
```   872   shows "integral\<^sup>N N f = (\<integral>\<^sup>+x. RN_deriv M N x * f x \<partial>M)"
```
```   873 proof -
```
```   874   have "integral\<^sup>N N f = integral\<^sup>N (density M (RN_deriv M N)) f"
```
```   875     using N by (simp add: density_RN_deriv)
```
```   876   also have "\<dots> = (\<integral>\<^sup>+x. RN_deriv M N x * f x \<partial>M)"
```
```   877     using f by (simp add: nn_integral_density)
```
```   878   finally show ?thesis by simp
```
```   879 qed
```
```   880
```
```   881 lemma null_setsD_AE: "N \<in> null_sets M \<Longrightarrow> AE x in M. x \<notin> N"
```
```   882   using AE_iff_null_sets[of N M] by auto
```
```   883
```
```   884 lemma (in sigma_finite_measure) RN_deriv_unique:
```
```   885   assumes f: "f \<in> borel_measurable M"
```
```   886   and eq: "density M f = N"
```
```   887   shows "AE x in M. f x = RN_deriv M N x"
```
```   888   unfolding eq[symmetric]
```
```   889   by (intro density_unique_iff[THEN iffD1] f borel_measurable_RN_deriv
```
```   890             density_RN_deriv_density[symmetric])
```
```   891
```
```   892 lemma RN_deriv_unique_sigma_finite:
```
```   893   assumes f: "f \<in> borel_measurable M"
```
```   894   and eq: "density M f = N" and fin: "sigma_finite_measure N"
```
```   895   shows "AE x in M. f x = RN_deriv M N x"
```
```   896   using fin unfolding eq[symmetric]
```
```   897   by (intro sigma_finite_density_unique[THEN iffD1] f borel_measurable_RN_deriv
```
```   898             density_RN_deriv_density[symmetric])
```
```   899
```
```   900 lemma (in sigma_finite_measure) RN_deriv_distr:
```
```   901   fixes T :: "'a \<Rightarrow> 'b"
```
```   902   assumes T: "T \<in> measurable M M'" and T': "T' \<in> measurable M' M"
```
```   903     and inv: "\<forall>x\<in>space M. T' (T x) = x"
```
```   904   and ac[simp]: "absolutely_continuous (distr M M' T) (distr N M' T)"
```
```   905   and N: "sets N = sets M"
```
```   906   shows "AE x in M. RN_deriv (distr M M' T) (distr N M' T) (T x) = RN_deriv M N x"
```
```   907 proof (rule RN_deriv_unique)
```
```   908   have [simp]: "sets N = sets M" by fact
```
```   909   note sets_eq_imp_space_eq[OF N, simp]
```
```   910   have measurable_N[simp]: "\<And>M'. measurable N M' = measurable M M'" by (auto simp: measurable_def)
```
```   911   { fix A assume "A \<in> sets M"
```
```   912     with inv T T' sets.sets_into_space[OF this]
```
```   913     have "T -` T' -` A \<inter> T -` space M' \<inter> space M = A"
```
```   914       by (auto simp: measurable_def) }
```
```   915   note eq = this[simp]
```
```   916   { fix A assume "A \<in> sets M"
```
```   917     with inv T T' sets.sets_into_space[OF this]
```
```   918     have "(T' \<circ> T) -` A \<inter> space M = A"
```
```   919       by (auto simp: measurable_def) }
```
```   920   note eq2 = this[simp]
```
```   921   let ?M' = "distr M M' T" and ?N' = "distr N M' T"
```
```   922   interpret M': sigma_finite_measure ?M'
```
```   923   proof
```
```   924     from sigma_finite_countable guess F .. note F = this
```
```   925     show "\<exists>A. countable A \<and> A \<subseteq> sets (distr M M' T) \<and> \<Union>A = space (distr M M' T) \<and> (\<forall>a\<in>A. emeasure (distr M M' T) a \<noteq> \<infinity>)"
```
```   926     proof (intro exI conjI ballI)
```
```   927       show *: "(\<lambda>A. T' -` A \<inter> space ?M') ` F \<subseteq> sets ?M'"
```
```   928         using F T' by (auto simp: measurable_def)
```
```   929       show "\<Union>((\<lambda>A. T' -` A \<inter> space ?M')`F) = space ?M'"
```
```   930         using F T'[THEN measurable_space] by (auto simp: set_eq_iff)
```
```   931     next
```
```   932       fix X assume "X \<in> (\<lambda>A. T' -` A \<inter> space ?M')`F"
```
```   933       then obtain A where [simp]: "X = T' -` A \<inter> space ?M'" and "A \<in> F" by auto
```
```   934       have "X \<in> sets M'" using F T' \<open>A\<in>F\<close> by auto
```
```   935       moreover
```
```   936       have Fi: "A \<in> sets M" using F \<open>A\<in>F\<close> by auto
```
```   937       ultimately show "emeasure ?M' X \<noteq> \<infinity>"
```
```   938         using F T T' \<open>A\<in>F\<close> by (simp add: emeasure_distr)
```
```   939     qed (insert F, auto)
```
```   940   qed
```
```   941   have "(RN_deriv ?M' ?N') \<circ> T \<in> borel_measurable M"
```
```   942     using T ac by measurable
```
```   943   then show "(\<lambda>x. RN_deriv ?M' ?N' (T x)) \<in> borel_measurable M"
```
```   944     by (simp add: comp_def)
```
```   945
```
```   946   have "N = distr N M (T' \<circ> T)"
```
```   947     by (subst measure_of_of_measure[of N, symmetric])
```
```   948        (auto simp add: distr_def sets.sigma_sets_eq intro!: measure_of_eq sets.space_closed)
```
```   949   also have "\<dots> = distr (distr N M' T) M T'"
```
```   950     using T T' by (simp add: distr_distr)
```
```   951   also have "\<dots> = distr (density (distr M M' T) (RN_deriv (distr M M' T) (distr N M' T))) M T'"
```
```   952     using ac by (simp add: M'.density_RN_deriv)
```
```   953   also have "\<dots> = density M (RN_deriv (distr M M' T) (distr N M' T) \<circ> T)"
```
```   954     by (simp add: distr_density_distr[OF T T', OF inv])
```
```   955   finally show "density M (\<lambda>x. RN_deriv (distr M M' T) (distr N M' T) (T x)) = N"
```
```   956     by (simp add: comp_def)
```
```   957 qed
```
```   958
```
```   959 lemma (in sigma_finite_measure) RN_deriv_finite:
```
```   960   assumes N: "sigma_finite_measure N" and ac: "absolutely_continuous M N" "sets N = sets M"
```
```   961   shows "AE x in M. RN_deriv M N x \<noteq> \<infinity>"
```
```   962 proof -
```
```   963   interpret N: sigma_finite_measure N by fact
```
```   964   from N show ?thesis
```
```   965     using sigma_finite_iff_density_finite[OF borel_measurable_RN_deriv, of N] density_RN_deriv[OF ac]
```
```   966     by simp
```
```   967 qed
```
```   968
```
```   969 lemma (in sigma_finite_measure)
```
```   970   assumes N: "sigma_finite_measure N" and ac: "absolutely_continuous M N" "sets N = sets M"
```
```   971     and f: "f \<in> borel_measurable M"
```
```   972   shows RN_deriv_integrable: "integrable N f \<longleftrightarrow>
```
```   973       integrable M (\<lambda>x. enn2real (RN_deriv M N x) * f x)" (is ?integrable)
```
```   974     and RN_deriv_integral: "integral\<^sup>L N f = (\<integral>x. enn2real (RN_deriv M N x) * f x \<partial>M)" (is ?integral)
```
```   975 proof -
```
```   976   note ac(2)[simp] and sets_eq_imp_space_eq[OF ac(2), simp]
```
```   977   interpret N: sigma_finite_measure N by fact
```
```   978
```
```   979   have eq: "density M (RN_deriv M N) = density M (\<lambda>x. enn2real (RN_deriv M N x))"
```
```   980   proof (rule density_cong)
```
```   981     from RN_deriv_finite[OF assms(1,2,3)]
```
```   982     show "AE x in M. RN_deriv M N x = ennreal (enn2real (RN_deriv M N x))"
```
```   983       by eventually_elim (auto simp: less_top)
```
```   984   qed (insert ac, auto)
```
```   985
```
```   986   show ?integrable
```
```   987     apply (subst density_RN_deriv[OF ac, symmetric])
```
```   988     unfolding eq
```
```   989     apply (intro integrable_real_density f AE_I2 enn2real_nonneg)
```
```   990     apply (insert ac, auto)
```
```   991     done
```
```   992
```
```   993   show ?integral
```
```   994     apply (subst density_RN_deriv[OF ac, symmetric])
```
```   995     unfolding eq
```
```   996     apply (intro integral_real_density f AE_I2 enn2real_nonneg)
```
```   997     apply (insert ac, auto)
```
```   998     done
```
```   999 qed
```
```  1000
```
```  1001 lemma (in sigma_finite_measure) real_RN_deriv:
```
```  1002   assumes "finite_measure N"
```
```  1003   assumes ac: "absolutely_continuous M N" "sets N = sets M"
```
```  1004   obtains D where "D \<in> borel_measurable M"
```
```  1005     and "AE x in M. RN_deriv M N x = ennreal (D x)"
```
```  1006     and "AE x in N. 0 < D x"
```
```  1007     and "\<And>x. 0 \<le> D x"
```
```  1008 proof
```
```  1009   interpret N: finite_measure N by fact
```
```  1010
```
```  1011   note RN = borel_measurable_RN_deriv density_RN_deriv[OF ac]
```
```  1012
```
```  1013   let ?RN = "\<lambda>t. {x \<in> space M. RN_deriv M N x = t}"
```
```  1014
```
```  1015   show "(\<lambda>x. enn2real (RN_deriv M N x)) \<in> borel_measurable M"
```
```  1016     using RN by auto
```
```  1017
```
```  1018   have "N (?RN \<infinity>) = (\<integral>\<^sup>+ x. RN_deriv M N x * indicator (?RN \<infinity>) x \<partial>M)"
```
```  1019     using RN(1) by (subst RN(2)[symmetric]) (auto simp: emeasure_density)
```
```  1020   also have "\<dots> = (\<integral>\<^sup>+ x. \<infinity> * indicator (?RN \<infinity>) x \<partial>M)"
```
```  1021     by (intro nn_integral_cong) (auto simp: indicator_def)
```
```  1022   also have "\<dots> = \<infinity> * emeasure M (?RN \<infinity>)"
```
```  1023     using RN by (intro nn_integral_cmult_indicator) auto
```
```  1024   finally have eq: "N (?RN \<infinity>) = \<infinity> * emeasure M (?RN \<infinity>)" .
```
```  1025   moreover
```
```  1026   have "emeasure M (?RN \<infinity>) = 0"
```
```  1027   proof (rule ccontr)
```
```  1028     assume "emeasure M {x \<in> space M. RN_deriv M N x = \<infinity>} \<noteq> 0"
```
```  1029     then have "0 < emeasure M {x \<in> space M. RN_deriv M N x = \<infinity>}"
```
```  1030       by (auto simp: zero_less_iff_neq_zero)
```
```  1031     with eq have "N (?RN \<infinity>) = \<infinity>" by (simp add: ennreal_mult_eq_top_iff)
```
```  1032     with N.emeasure_finite[of "?RN \<infinity>"] RN show False by auto
```
```  1033   qed
```
```  1034   ultimately have "AE x in M. RN_deriv M N x < \<infinity>"
```
```  1035     using RN by (intro AE_iff_measurable[THEN iffD2]) (auto simp: less_top[symmetric])
```
```  1036   then show "AE x in M. RN_deriv M N x = ennreal (enn2real (RN_deriv M N x))"
```
```  1037     by auto
```
```  1038   then have eq: "AE x in N. RN_deriv M N x = ennreal (enn2real (RN_deriv M N x))"
```
```  1039     using ac absolutely_continuous_AE by auto
```
```  1040
```
```  1041
```
```  1042   have "N (?RN 0) = (\<integral>\<^sup>+ x. RN_deriv M N x * indicator (?RN 0) x \<partial>M)"
```
```  1043     by (subst RN(2)[symmetric]) (auto simp: emeasure_density)
```
```  1044   also have "\<dots> = (\<integral>\<^sup>+ x. 0 \<partial>M)"
```
```  1045     by (intro nn_integral_cong) (auto simp: indicator_def)
```
```  1046   finally have "AE x in N. RN_deriv M N x \<noteq> 0"
```
```  1047     using RN by (subst AE_iff_measurable[OF _ refl]) (auto simp: ac cong: sets_eq_imp_space_eq)
```
```  1048   with eq show "AE x in N. 0 < enn2real (RN_deriv M N x)"
```
```  1049     by (auto simp: enn2real_positive_iff less_top[symmetric] zero_less_iff_neq_zero)
```
```  1050 qed (rule enn2real_nonneg)
```
```  1051
```
```  1052 lemma (in sigma_finite_measure) RN_deriv_singleton:
```
```  1053   assumes ac: "absolutely_continuous M N" "sets N = sets M"
```
```  1054   and x: "{x} \<in> sets M"
```
```  1055   shows "N {x} = RN_deriv M N x * emeasure M {x}"
```
```  1056 proof -
```
```  1057   from \<open>{x} \<in> sets M\<close>
```
```  1058   have "density M (RN_deriv M N) {x} = (\<integral>\<^sup>+w. RN_deriv M N x * indicator {x} w \<partial>M)"
```
```  1059     by (auto simp: indicator_def emeasure_density intro!: nn_integral_cong)
```
```  1060   with x density_RN_deriv[OF ac] show ?thesis
```
```  1061     by (auto simp: max_def)
```
```  1062 qed
```
```  1063
```
```  1064 end
```